![]() | ||
Parameters μ {\displaystyle \mu } real κ > 0 {\displaystyle \kappa >0} Support x ∈ {\displaystyle x\in } any interval of length 2π PDF e κ cos ( x − μ ) 2 π I 0 ( κ ) {\displaystyle {\frac {e^{\kappa \cos(x-\mu )}}{2\pi I_{0}(\kappa )}}} CDF (not analytic – see text) Mean μ {\displaystyle \mu } Median μ {\displaystyle \mu } |
In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle
Contents
Definition
The von Mises probability density function for the angle x is given by:
where I0(k) is the modified Bessel function of order 0.
The parameters μ and 1/κ are analogous to μ and σ2 (the mean and variance) in the normal distribution:
The probability density can be expressed as a series of Bessel functions (see Abramowitz and Stegun §9.6.34)
where Ij(x) is the modified Bessel function of order j.
The cumulative distribution function is not analytic and is best found by integrating the above series. The indefinite integral of the probability density is:
The cumulative distribution function will be a function of the lower limit of integration x0:
Moments
The moments of the von Mises distribution are usually calculated as the moments of the complex exponential z = eix rather than the angle x itself. These moments are referred to as circular moments. The variance calculated from these moments is referred to as the circular variance. The one exception to this is that the "mean" usually refers to the argument of the complex mean.
The nth raw moment of z is:
where the integral is over any interval
The mean of the complex exponential z is then just
and the circular mean value of the angle x is then taken to be the argument μ. This is the expected or preferred direction of the angular random variables. The variance of z, or the circular variance of x is:
Limiting behavior
In the limit of large κ the distribution becomes a normal distribution
where σ2 = 1/κ. In the limit of small κ it becomes a uniform distribution:
where the interval for the uniform distribution U(x) is the chosen interval of length 2π.
Estimation of parameters
A series of N measurements
and its expectation value will be just the first moment:
In other words,
Viewing the
and its expectation value is:
In other words, the statistic
will be an unbiased estimator of
Distribution of the mean
The distribution of the sample mean
where N is the number of measurements and
and
Note that product term in parentheses is just the distribution of the mean for a circular uniform distribution.
This means that the distribution of the mean direction
Entropy
The information entropy of the Von Mises distribution is defined as:
where
The characteristic function representation for the Von Mises distribution is:
where
For
Notice that the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified or, equivalently, the circular mean and circular variance are specified.